PlanetMath: section of a fiber bundle

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section of a fiber bundle

(Definition)

Let $p:E\rightarrow B$ be a fiber bundle, denoted by $\xi.$

A section of $\xi$ is a continuous map $s:B\rightarrow E$ such that the composition $p\circ s$ equals the identity. That is, for every $b\in B,$ is an element of the fiber over More generally, given a topological subspace of a section of $\xi$ over is a section of the restricted bundle ${p}\vert _{A}:p^{-1}(A)\rightarrow A.$

The set of sections of $\xi$ over is often denoted by $\Gamma(A;\xi),$ or by $\Gamma(\xi)$ for sections defined on all of Elements of $\Gamma(\xi)$ are sometimes called global sections, in contrast with the local sections $\Gamma(U;\xi)$ defined on an open set

Remark 1 If

and

have, for example, smooth structures, one can talk about smooth sections of the bundle. According to the context, the notation $\Gamma(\xi)$ often denotes smooth sections, or some other set of suitably restricted sections.

Example 1 If $\xi$ is a trivial fiber bundle with fiber

so that $E=F\times B$ and

is projection to

then sections of $\xi$ are in a natural bijective correspondence with continuous functions $B\rightarrow F.$

Example 2 If

is a smooth manifold and

its tangent bundle, a (smooth) section of this bundle is precisely a (smooth) tangent vector field.

In fact, any tensor field on a smooth manifold is a section of an appropriate vector bundle. For instance, a contravariant -tensor field is a section of the bundle $TM^{\otimes k}$ obtained by repeated tensor product from the tangent bundle, and similarly for covariant and mixed tensor fields.

Example 3 If

is a smooth manifold which is smoothly embedded in a Riemannian manifold

we can let the fiber over $b\in B$ be the orthogonal complement in

of the tangent space

. These choices of fiber turn out to make up a vector bundle $\nu(B)$ over

called the normal bundle of

. A section of $\nu(B)$ is a normal vector field on

Example 4 If $\xi$ is a vector bundle, the zero section is defined simply by

the zero vector on the fiber.

It is interesting to ask if a vector bundle admits a section which is nowhere zero. The answer is yes, for example, in the case of a trivial vector bundle, but in general it depends on the topology of the spaces involved. A well-known case of this question is the hairy ball theorem, which says that there are no nonvanishing tangent vector fields on the sphere.